0,3

This is a variant of the following property of the Catalan sequence:

A000108(n) = sum of the squares of the coefficients of x^(2k) in Catalan(x^2)^{n-2k+1}, as k varies from 0 to floor(n/2) where Catalan(x) = g.f. of A000108.

a(n) = Sum_{k=0..n\2} ( [x^(2*k)] A(x^2)^{2*(n-2*k)+1} )^2 for n>0 with a(0)=1.

To illustrate the recurrence, list coefficients of A(x^2)^(2n+1):

A^1: . 1,... 1,... 2,... 10,... 30,... 131,.......;

A^3: .... 1,... 3,... 9,... 43,... 168,... 735, ...;

A^5: ....... 1,... 5,... 20,... 100,... 455,.......;

A^7: .......... 1,... 7,... 35,... 189,... 959, ...;

A^9: ............. 1,... 9,... 54,... 318,.......;

A^11: ............... 1,... 11,... 77,... 495, ...;

A^13: .................. 1,... 13,... 104,.......;

A^15: ..................... 1,... 15,... 135, ...;

A^17: ........................ 1,... 17,.......;

A^19: ........................... 1,... 19, ...;

A^21: .............................. 1,.......;

A^23: ................................. 1, ...;...

then sum the squares of the coefficients in each column:

a(0) = 1^2 = 1;

a(1) = 1^2 = 1;

a(2) = 1^2 + 1^2 = 2;

a(3) = 3^2 + 1^2 = 10;

a(4) = 2^2 + 5^2 + 1^2 = 30;

a(5) = 9^2 + 7^2 + 1^2 = 131;

a(6) = 10^2 + 20^2 + 8^2 + 1^2 = 582;

a(7) = 43^2 + 35^2 + 11^2 + 1^2 = 3196;

a(8) = 30^2 + 100^2 + 54^2 + 13^2 + 1^2 = 13986;

a(9) = 168^2 + 189^2 + 77^2 + 15^2 + 1^2 = 70100.

(PARI) {a(n)=local(A=1+sum(j=1, n\2, a(j)*x^(2*j))+x*O(x^n)); if(n==0, 1, sum(k=0, n\2, polcoeff(A^(2*(n-2*k)+1), 2*k)^2))}

Cf. A095892.

Sequence in context: A058967 A140576 A064932 this_sequence A156492 A090809 A051747

Adjacent sequences: A162246 A162247 A162248 this_sequence A162250 A162251 A162252

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Jun 28 2009

Comment corrected by Paul D. Hanna (pauldhanna(AT)juno.com), Jul 05 2009